We report on the proton spin-lattice relaxation measurements in the isotropic and nematic phases of the liquid crystal 4’-pentyl-4-cyanobiphenyl (5CB), constrained by a low-concentration polymer network. Above the nematic-isotropic transition, a relatively small impact of the network in the MHz regime and a huge increase in the relaxation rate in the kHz regime are explained consistently by processes in the weakly orientationally ordered surface layer. A dynamic process with the correlation time similar to 4X10(-6) s is detected. The orientational order parameter in the surface layer is found to be almost temperature independent, and is estimated as S-0 similar to 0.1. The large internal surface-to-volume ratio of the liquid crystal in the dispersion, required for the explanation of experimental data, supports the simplified model which envisions the network elements as tiny fibers with nanometer radius. Comparison between. NMR and optical data reveals mutual cross linking of the fibers. Measurements of the molecular self-diffusion constant indicate that fibers are distributed inhomogeneously on the submicrometer scale.